If you had any number of ordinary dice, what are the possible ways
of making their totals 6? What would the product of the dice be
each time?

The Deca Tree

Stage: 2 Challenge Level:

The Deca Tree

In the forest there is a Deca Tree.
A Deca Tree has $10$ trunks,

and on each trunk there are $10$ branches,

and on each branch there are $10$ twigs,

and on each twig there are $10$ leaves:

One day a woodcutter came along and cut down one trunk from the tree.
Then he cut off one branch from another trunk of the tree.
Then he cut off one twig from another branch.
Finally he pulled one leaf from another twig.

Why do this problem?

This problem can give some substance to children's ideas on place value especially when you are introducing or using numbers greater than $1000$. It shows in a pictorial way what the extra zeros do to a number when multiplying by $10$ and $100$.

The problem can also help children to appreciate how big large numbers really are!

Possible approach

You could introduce the problem by telling the class the 'story' about the tree and its trunks, its branches, its twig and its leaves. During, or after this, ask such questions as "How many branches are there altogether?" and "How many leaves will there be on each branch?".

Then, working in pairs, the children could work out the details of the problem. It might help them to draw rough sketches and/or make a table/list of the numbers of the trunks, branches, twigs and leaves, both at the beginning and after the woodcutter has been at work.

After the number of leaves taken off the tree has been calculated this needs to be subtracted from the total of all the leaves on the tree. This might be the time for a plenary so that you work on the calculation together as as class.

Some children may have do the problem differently, such as deducting the leaves removed from the tree as they go. This could lead to an interesting discussion on different methods to tackling problems such as this.

Key questions

If there are ten twigs on each branch, how many will there be on ten branches?

How many leaves are there on ten twigs?

How many leaves did the Deca Tree have before the woodcutter came along?

How many leaves did the woodcutter chop off each time?

Possible extension

Children could be invited to invent a similar tree with different numbers of branches etc. such a five-branched 'Penta' tree or an eight-branched 'Octa' tree.

Possible support

In addition to, or instead of, drawing sketches, it might help some children to make a model of the tree, or at least part of the tree. Pipecleaners could work well for this purpose.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.